On the minimization over SO(3) Manifolds

Transcription

1 On the minimization over SO(3) Manifolds Frank O. Kuehnel RIACS NASA Ames Research Center, MS 69-, Moffett Field, CA 9435, USA Abstract. In almost all image-model and model-model registration problems the question arises as to what optimal rigid body transformation applies to bring a physical 3-dimensional model in alignment with the observed one. Data may also be corrupted by noise. Here I will present the exponential and quaternion representations for the SO(3) group. I will present the technique of compounding derivatives and demonstrate that it is most suited for dealing with numerical optimization problems that involve rotation groups. 1 Introduction A common task in computer vision is matching images or features and estimating essential transformation parameters [1]. In the weak perspective regime the -dimensional affine image transformation with 6 parameters is applicable; in general a perspective transformation applies. Amongst Fig. 1. Fly-by satellite view on a rendered region of duckwater. The main difference between both images is a rotation about the viewing axis. those parameters are 3 Euler-angles that describe the orientation of the

2 viewer with respect to a world-coordinate system, see Fig. 1 as an example. Then neglecting other free parameters, we could formulate the image matching problem as finding a minimum to the log- likelihood log P p (I p Îp(φ,θ,ψ)), (1) where we sum over all pixels in the observed image data I p and the expected image Îp. It is well known that numerical optimiztion algorithms with Euler-angle representation have numerical problems. Several alternatives to Euler-angle representations exist. However, a prior it is not clear how well various representations and minimization methods will perfom. The subject of my investigation is the types of suitable rotation group representations and their application in numerical minimization algorithms. I will compare the convergence rate of various methods in the special case of matching point-clouds. In the next section, I will present suitable rotation group representations, followed by their application within numerical optimization algorithms. SO(3) reprsentations I briefly introduce the euler, the exponential and the quaternion representation. An extensive introduction to rotation groups and parametrization can be found in []..1 Euler-angle representation The rotation matrix R represents an orthonormal transformation, RR T = I, det(r) = 1, as such it can be decomposed into simpler rotation matrices, R = R z (φ) R y (θ)r z (ψ), () with cos φ sinφ cos θ sin θ R z (φ) = sinφ cos φ, R y (θ) = 1. (3) 1 sinθ cos θ The decomposition () is not unique. We adhere to the NASA Standard Aerospace convention [3] with ψ the precession, θ the nutation and φ as the spin. The derivatives with respect to the Euler-angles are easily obtained and will not be given here.

8 with rotation matrix components occuring at most in order, hence we could call them R µν -harmonic functions, it generally doesn t possess a closed form solution. Only for the particular case presented below the solution is known. 3.1 Pseudo-quadratic functions The special case of finding a unit quaternion ˆQ min which minimizes f s (R( ˆQ)) N = a + Rc i d i (35) i=1 can be solved analytically [6, 7]. (35) represents the case where a cloud of N points c i in IR 3 are mapped by a rotation such that the result most closely resembles the cloud d i. We can write (35) in tensor notation, f s = a + d µ d µ d µ c ν R µν + c ν c ρ δ µη R µν R ηρ, (36) where we have omitted the sum over point index i, thus in terms of tensor coefficients (34) we have b µν = c µ d ν, Now, we use the orthogonality relation, c ηρ µν = c νc ρ δ µη. c ηρ µν R µνr ηρ = c ν c ρ δ νρ (37) and obtain an expression that is linear in the matrix elements R µν, hence (35) constitutes only a pseudo-quadratic function, n f s = a + b µν R µν, b µν = d i c T i, a = a + c T i c i + d T i d i. (38) i=1 According to [7] we can find a solution to the R µν linear problem (38) by simply decomposing the general rank- tensor b µν and restating the problem as in (35). Then we solve for ˆQ min by noticing that f s is quadratic in the quaternion components and therefore can be written as n f s = a + (q,q) Bi T B i(q,q) T, (39) i=1 with 4 4 matricies B i, [ ] (ci d B i = i ) T. d i c i [d i + c i ] A solution to (35) is given by the eigenvector ˆq of B T i B i with minimal eigenvalue.

12 and employ the minimization methods. The graphs show the minimization trajectory in the Rodrigues vector space. The green dota represents the initial guess, the end-points of the two red lines the two possible exact solutions. We track the convergence by counting the number of steps it takes to obtain the accuracy f s. This corresponds to the residual value of mismatching the euclidian distance (35). We clearly notice that in all scenarios the compounding quaternion derivative method is vastly superior and has super-linear convergence. Also it never failed in all tested cases. Most rarely does the Euler-angle method succeed. 6 Conclusions I have concisely introduced important rotation group SO(3) representations and presented their differential structure. I ve developed the new idea of a product (compounding) path-quaternions and demonstrated its advantages application towards numerical minimization methods. In the special point-matching case this method is vastly superior than all its alternatives. It is expected that the super-linear convergence of the compounding path-quaternion approach is retained even in the case of more general function types that depend on SO(3) parameters.

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